3.3 Calculating the Final Payment

Introduction

If you have ever paid off a loan you may have noticed that your last payment was a slightly different amount than your other payments. Whether you are making monthly insurance premium payments, paying municipal property tax instalments, financing your vehicle, paying your mortgage, receiving monies from an investment annuity, or dealing with any other situation where an annuity is extinguished through equal payments, the last payment typically differs from the rest, by as little as one penny or up to a few dollars. This difference can be much larger if you arbitrarily chose an annuity payment as opposed to determining an accurate payment through time value of money calculations.

Why is it important for this final payment to differ from all of the previous payments? From a consumer perspective, you do not want to pay a cent more toward a debt than you have to. In [latex]2011[/latex], the average Canadian is more than [latex]\$100,000[/latex] in debt across various financial tools such as car loans, consumer debt, and mortgages. Imagine if you overpaid every one of those debts by a dollar. Over the course of your lifetime those overpayments would add up to hundreds or even thousands of dollars.

Calculating the Final Payment

When you calculate out the payment for a loan, the actual payment is rounded to two decimal places. It is rare for a calculated loan payment not to require rounding. The rounding up or down of the payment forms the basis for adjusting the final payment.

Consider the situation where the loan payment is rounded up. Suppose the calculated loan payment is [latex]\$999.995[/latex]. Rounding to two decimals, the actual payment made is [latex]\$1,000[/latex]. So with each [latex]\$1,000[/latex] loan payment, you are overpaying the debt by [latex]\$1,000− \$999.995 = \$0.005[/latex]. If you make [latex]20[/latex] such payments, you end up overpaying the debt by [latex]20\times \$0.005=\$0.10[/latex]. Therefore, when it comes to the final payment you need to compensate for all of the overpayments made, reducing the final payment by [latex]\$0.10[/latex]. Because the principal is slightly smaller at all times as a result of the overpayment, an additional adjustment may be needed because of less interest being calculated.

Now, consider the situation where the loan payment is rounded down. Suppose the calculated loan payment is [latex]\$1,000.0025[/latex]. Rounding to two decimals, the actual payment made is [latex]\$1,000[/latex]. So, with each [latex]\$1,000[/latex] loan payment, you are underpaying the debt by [latex]\$1,000.0025-\$1,000=\$0.0025[/latex]. If you make [latex]20[/latex] such payments, you end up underpaying by [latex]20\times \$0.0025=\$0.05[/latex]. Therefore, when it comes to the final payment, you need to increase the final payment by this amount. Because the principal is slightly larger at all times as a result of the underpayment, an additional adjustment may be needed because of more interest being calculated.

We have already learned how to calculate the final payment by completing the last row of the amortization schedule. This method gives the following formula for calculating the final payment.

[latex]\mbox{Final Payment} = \mbox{Balance from Second Last Row}+\mbox{Interest from Last Row}[/latex]

Alternatively, we can calculate the final payment without constructing the last row of the amortization schedule. This method is based on the assumption that all of the payments, including the final payment are the same. The final payment is found by adjusting the regular payment ([latex]PMT[/latex]) by the amount overpaid or underpaid.

[latex]\begin{eqnarray*} \mbox{Final Payment} & = & PMT-\mbox{Amount Overpaid} \\ \\ & \mbox{or} & \\ \\ \mbox{Final Payment} & = & PMT+\mbox{Amount Underpaid} \end{eqnarray*}[/latex]